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A316833 Sums of four distinct odd squares. 3
84, 116, 140, 156, 164, 180, 196, 204, 212, 228, 236, 244, 252, 260, 276, 284, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 396, 404, 420, 428, 436, 444, 452, 460, 468, 476, 484, 492, 500, 508, 516, 524, 532, 540, 548, 556, 564, 572, 580, 588, 596, 604, 612, 620, 628, 636, 644, 652, 660 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Theorem (Conjectured by R. William Gosper, proved by M. D. Hirschhorn): Any sum of four distinct odd squares is the sum of four distinct even squares.

The proof uses the following identity:

(4a+1)^2+(4b+1)^2+(4c+1)^2+(4d+1)^2 = 4[ (a+b+c+d+1)^2 + (a-b-c+d)^2 + (a-b+c-d)^2 + (a+b-c-d)^2 ].

All terms == 4 (mod 8).  Are all numbers == 4 (mod 8) and > 412 members of the sequence? - Robert Israel, Jul 20 2018

REFERENCES

R. William Gosper and Stephen K. Lucas, Postings to Math Fun Mailing List, July 19 2018

Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

MAPLE

N:= 1000: # to get all terms <= N

V:= Vector(N):

for a from 1 to floor(sqrt(N/4)) by 2 do

  for b from a+2 to floor(sqrt((N-a^2)/3)) by 2 do

    for c from b+2 to floor(sqrt((N-a^2-b^2)/2)) by 2 do

      for d from c + 2  by 2 do

        r:= a^2+b^2+c^2+d^2;

        if r > N then break fi;

        V[r]:= V[r]+1

od od od od:

select(t -> V[t]>=1, [$1..N]); # Robert Israel, Jul 20 2018

CROSSREFS

A316834 lists the subsequence for which the representation is unique.

Cf. A004433, A316835.

Sequence in context: A157119 A209204 A219801 * A316834 A227734 A192322

Adjacent sequences:  A316830 A316831 A316832 * A316834 A316835 A316836

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jul 19 2018

STATUS

approved

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Last modified June 6 08:44 EDT 2020. Contains 334859 sequences. (Running on oeis4.)